When drilling a wellbore at or nearly balanced, an influx of fluid into the wellbore from a formation intersected by the open hole can be experienced. It is common practice to stop drilling and shut in a well (close the blowout preventers and stop circulating) when undesired influxes are experienced. There are several well known procedures for dealing with large influxes (such as, the driller's method, the weight and wait method, etc.). However, these methods commonly rely on circulating the influx out of the wellbore through the rig's choke and manifold, with the choke being typically hydraulically actuated (but manually controlled) and incapable of responding quickly and in fine incremental steps to changing conditions to maintain a desired bottomhole pressure.